# Power of a Power Rule

## Formula

$\large {(b^{\displaystyle m})}^{\displaystyle n} = b^{\displaystyle mn}$

### Proof

$b$ is a literal number and it is multiplied to same literal $m$ times. The quantity of the product of them is written as $b^{\displaystyle m}$ in exponential notation.

$b^{\displaystyle m} = \underbrace{b \times b \times b \times \ldots \times b}_{\displaystyle m \, factors}$

Now, multiply $b^{\displaystyle m}$ by same term $n$ times and the product of them is written as ${(b^{\displaystyle m})}^{\displaystyle n}$ in exponential form.

${(b^{\displaystyle m})}^{\displaystyle n} = \underbrace{b^{\displaystyle m} \times b^{\displaystyle m} \times b^{\displaystyle m} \times \ldots \times b^{\displaystyle m}}_{\displaystyle n \, factors}$

Express each exponential term $b^{\displaystyle m}$ in its product form.

${(b^{\displaystyle m})}^{\displaystyle n} = \underbrace{\underbrace{(b \times b \times b \times \ldots \times b)}_{\displaystyle m \, factors} \times \underbrace{(b \times b \times b \times \ldots \times b)}_{\displaystyle m \, factors} \times \underbrace{(b \times b \times b \times \ldots \times b)}_{\displaystyle m \, factors} \times \ldots \times \underbrace{(b \times b \times b \times \ldots \times b)}_{\displaystyle m \, factors}}_{\displaystyle n \, factors}$

Each $b^{\displaystyle m}$ term contains $m$ multiplying factors of $b$. So, $n$ terms contains $n \times m$ multiplying factors. Therefore, the number of multiplying factors of $b$ is $mn$ in the product.

${(b^{\displaystyle m})}^{\displaystyle n} = \underbrace{b \times b \times b \times \ldots \times b}_{\displaystyle m \times n \, factors}$

$\implies {(b^{\displaystyle m})}^{\displaystyle n} = \underbrace{b \times b \times b \times \ldots \times b}_{\displaystyle mn \, factors}$

The literal quantity $b$ is multiplied by itself $mn$ times. So, it can be expressed in exponential form.

${(b^{\displaystyle m})}^{\displaystyle n} = b^{\displaystyle mn}$

The property expresses that the exponent of an exponential term is equal to the base number raised the power of product of the exponents. It is called the exponent product rule of an exponential term.

#### Verification

${(2^3)}^5$ is an example for the exponent of an exponential term.

$\implies$ ${(2^3)}^5 \,=\, {(2 \times 2 \times 2)}^5$

$\implies$ ${(2^3)}^5 \,=\, 8^5$

$\implies$ ${(2^3)}^5 \,=\, 8 \times 8 \times 8 \times 8 \times 8$

$\implies$ ${(2^3)}^5 \,=\, 32768$

Now, find the value of $2$ raised the power of the product of exponents $3$ and $5$.

$2^{3 \times 5} = 2^{15}$

$\implies$ $2^{3 \times 5}$ $=$ $2$ $\times$ $2$ $\times$ $2$ $\times$ $2$ $\times$ $2$ $\times$ $2$ $\times$ $2$ $\times$ $2$ $\times$ $2$ $\times$ $2$ $\times$ $2$ $\times$ $2$ $\times$ $2$ $\times$ $2$ $\times$ $2$

$\implies$ $2^{3 \times 5} \,=\, 32768$

Compare both values and they both same.

$\therefore \,\,\,\,\,\,$ ${(2^3)}^5 \,=\, 2^{3 \times 5} \,=\, 32768$